In that paper, Garland proved a conjecture of Serre, that the cohomology of co-compact lattices in high-rank simple \(p\) -adic groups vanishes. Equivalently, if \(X\)  is a finite simplicial quotient of a Bruhat-Tits building of dimension at least two, its cohomology vanishes. The proof of Garland is “local-to-global”: he showed that if the links of relevant cells have a spectral gap, then so does the global Laplacian of \(X\) . Namely, if \(X\)  is locally an expander, then it is also globally so. (For a purely combinatorial treatment and generalizations - see [1]}). The global spectral gap implies the vanishing of the cohomology.
